Find the Roots (Zeros) 2x^4+x^3-x^2-x-1

$2 x^{4} + x^{3} - x^{2} - x - 1$

Set $2 x^{4} + x^{3} - x^{2} - x - 1$ equal to $0$ .

$2 x^{4} + x^{3} - x^{2} - x - 1 = 0$

Solve for $x$ .

Factor the left side of the equation.

Regroup terms.

$x^{3} - x^{2} + 2 x^{4} - x - 1 = 0$

Factor $x^{2}$ out of $x^{3} - x^{2}$ .

Factor $x^{2}$ out of $x^{3}$ .

$x^{2} x - x^{2} + 2 x^{4} - x - 1 = 0$

Factor $x^{2}$ out of $- x^{2}$ .

$x^{2} x + x^{2} \cdot - 1 + 2 x^{4} - x - 1 = 0$

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 1$ .

$x^{2} (x - 1) + 2 x^{4} - x - 1 = 0$

Factor $2 x^{4} - x - 1$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1 q = \pm 1, \pm 2$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.5$

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

Substitute $1$ into the polynomial.

$2 \cdot 1^{4} - 1 \cdot 1 - 1$

Raise $1$ to the power of $4$ .

$2 \cdot 1 - 1 \cdot 1 - 1$

Multiply $2$ by $1$ .

$2 - 1 \cdot 1 - 1$

Multiply $- 1$ by $1$ .

$2 - 1 - 1$

Subtract $1$ from $2$ .

$1 - 1$

Subtract $1$ from $1$ .

$0$

Since $1$ is a known root, divide the polynomial by $x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{4} - x - 1}{x - 1}$

Divide $2 x^{4} - x - 1$ by $x - 1$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

Divide the highest order term in the dividend $2 x^{4}$ by the highest order term in divisor $x$ .

$2 x^{3}$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} - 2 x^{3}$

$2 x^{3}$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$2 x^{3}$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

Pull the next terms from the original dividend down into the current dividend.

$2 x^{3}$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

$2 x^{3}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 2 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$2 x^{3}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

$2 x^{3}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

Divide the highest order term in the dividend $2 x^{2}$ by the highest order term in divisor $x$ .

$2 x^{3}$

$2 x^{2}$

$2 x$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$2 x^{2}$

$2 x$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$2 x$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - 2 x$

$2 x^{3}$

$2 x^{2}$

$2 x$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$2 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$2 x^{3}$

$2 x^{2}$

$2 x$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$2 x$

$x$

Pull the next terms from the original dividend down into the current dividend.

$2 x^{3}$

$2 x^{2}$

$2 x$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$2 x$

$x$

$1$

Divide the highest order term in the dividend $x$ by the highest order term in divisor $x$ .

$2 x^{3}$

$2 x^{2}$

$2 x$

$1$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$2 x$

$x$

$1$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$2 x^{2}$

$2 x$

$1$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$2 x$

$x$

$1$

$x$

$1$

The expression needs to be subtracted from the dividend, so change all the signs in $x - 1$

$2 x^{3}$

$2 x^{2}$

$2 x$

$1$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$2 x$

$x$

$1$

$x$

$1$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$2 x^{3}$

$2 x^{2}$

$2 x$

$1$

$x$

$1$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$2 x$

$x$

$1$

$x$

$1$

$0$

Since the remander is $0$ , the final answer is the quotient.

$2 x^{3} + 2 x^{2} + 2 x + 1$

Write $2 x^{4} - x - 1$ as a set of factors.

$x^{2} (x - 1) + (x - 1) (2 x^{3} + 2 x^{2} + 2 x + 1) = 0$

Factor $x - 1$ out of $x^{2} (x - 1) + (x - 1) (2 x^{3} + 2 x^{2} + 2 x + 1)$ .

Factor $x - 1$ out of $x^{2} (x - 1)$ .

$(x - 1) x^{2} + (x - 1) (2 x^{3} + 2 x^{2} + 2 x + 1) = 0$

Factor $x - 1$ out of $(x - 1) x^{2} + (x - 1) (2 x^{3} + 2 x^{2} + 2 x + 1)$ .

$(x - 1) (x^{2} + 2 x^{3} + 2 x^{2} + 2 x + 1) = 0$

Add $x^{2}$ and $2 x^{2}$ .

$(x - 1) (2 x^{3} + 3 x^{2} + 2 x + 1) = 0$

Factor.

Factor $2 x^{3} + 3 x^{2} + 2 x + 1$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1 q = \pm 1, \pm 2$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.5$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

$2 {(- 1)}^{3} + 3 {(- 1)}^{2} + 2 \cdot - 1 + 1$

Raise $- 1$ to the power of $3$ .

$2 \cdot - 1 + 3 {(- 1)}^{2} + 2 \cdot - 1 + 1$

Multiply $2$ by $- 1$ .

$- 2 + 3 {(- 1)}^{2} + 2 \cdot - 1 + 1$

Raise $- 1$ to the power of $2$ .

$- 2 + 3 \cdot 1 + 2 \cdot - 1 + 1$

Multiply $3$ by $1$ .

$- 2 + 3 + 2 \cdot - 1 + 1$

Add $- 2$ and $3$ .

$1 + 2 \cdot - 1 + 1$

Multiply $2$ by $- 1$ .

$1 - 2 + 1$

Subtract $2$ from $1$ .

$- 1 + 1$

Add $- 1$ and $1$ .

$0$

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} + 3 x^{2} + 2 x + 1}{x + 1}$

Divide $2 x^{3} + 3 x^{2} + 2 x + 1$ by $x + 1$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$2 x^{3}$

$3 x^{2}$

$2 x$

$1$

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

					$2 x^{2}$
	$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			+	$2 x^{3}$	+	$2 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 2 x^{2}$

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	+	$2 x$

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$x$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	+	$2 x$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	+	$2 x$
					+	$x^{2}$	+	$x$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + x$

				$2 x^{2}$	+	$x$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	+	$2 x$
					-	$x^{2}$	-	$x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	+	$x$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	+	$2 x$
					-	$x^{2}$	-	$x$
							+	$x$

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$	+	$x$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	+	$2 x$
					-	$x^{2}$	-	$x$
							+	$x$	+	$1$

Divide the highest order term in the dividend $x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$x$	+	$1$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	+	$2 x$
					-	$x^{2}$	-	$x$
							+	$x$	+	$1$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$	+	$1$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	+	$2 x$
					-	$x^{2}$	-	$x$
							+	$x$	+	$1$
							+	$x$	+	$1$

The expression needs to be subtracted from the dividend, so change all the signs in $x + 1$

				$2 x^{2}$	+	$x$	+	$1$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	+	$2 x$
					-	$x^{2}$	-	$x$
							+	$x$	+	$1$
							-	$x$	-	$1$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	+	$x$	+	$1$
$x$	+	$1$		$2 x^{3}$	+	$3 x^{2}$	+	$2 x$	+	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	+	$2 x$
					-	$x^{2}$	-	$x$
							+	$x$	+	$1$
							-	$x$	-	$1$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$2 x^{2} + x + 1$

Write $2 x^{3} + 3 x^{2} + 2 x + 1$ as a set of factors.

$(x - 1) ((x + 1) (2 x^{2} + x + 1)) = 0$

Remove unnecessary parentheses.

$(x - 1) (x + 1) (2 x^{2} + x + 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$x + 1 = 0$

$2 x^{2} + x + 1 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $2 x^{2} + x + 1$ equal to $0$ and solve for $x$ .

Set $2 x^{2} + x + 1$ equal to $0$ .

$2 x^{2} + x + 1 = 0$

Solve $2 x^{2} + x + 1 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 2$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (2 \cdot 1)}}{2 \cdot 2}$

Simplify.

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot 1}}{2 \cdot 2}$

Multiply $- 8$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 8}}{2 \cdot 2}$

Subtract $8$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 7}}{2 \cdot 2}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 2}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm i \sqrt{7}}{4}$

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{7}}{4}, - \frac{1 + i \sqrt{7}}{4}$

The final solution is all the values that make $(x - 1) (x + 1) (2 x^{2} + x + 1) = 0$ true.

$x = 1, - 1, - \frac{1 - i \sqrt{7}}{4}, - \frac{1 + i \sqrt{7}}{4}$

Find the Roots (Zeros) 2x^4+x^3-x^2-x-1

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